Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites

Abstract

a b s t r a c tIt is well-known that classical continuum theory has certain deficiencies in predicting material's behavior at the micro-and nanoscales, where the size effect is not negligible. Higher order continuum theories introduce new material constants into the formulation, making the interpretation of the size effect possible. One famous version of these theories is the couple stress theory, invoked to study the anti-plane problems of the elliptic inhomogeneities and inclusions in the present work. The formulation in elliptic coordinates leads to an exact series solution involving Mathieu functions. Subsequently, the elastic fields of a single inhomogeneity in conjunction with the Mori-Tanaka theory is employed to estimate the overall anti-plane shear moduli of composites with uni-directional elliptic cylindrical fibers. The dependence of the anti-plane elastic moduli on several important physical parameters such as size, aspect ratio and rigidity of the fiber, the characteristic length of the constituents, and the orientation of the reinforcements is analyzed. Based on the available data in the literature, certain nano-composite models have been proposed and their overall behavior estimated using the present theory.

Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

“…Anti-plane strain inclusion problems are simpler than plane strain and other 2-D and 3-D inclusion problems and have been extensively studied (e.g., Pak, 1992;Gao, 1996;Gao and Li, 2005;Lubarda, 2003;Le Quang et al, 2008;Haftbaradaran and Shodja, 2009). In particular, it was shown in Le Quang et al (2008) that within the context of classical elasticity the Eshelby tensor for the infinite-domain anti-plane strain inclusion problem is a second-order tensor (rather than a fourth-order tensor as in plane strain and other 2-D and 3-D inclusion problems).…”

Effective anti-plane moduli of couple stress composites containing elliptic multi-coated nano-fibers with interfacial damage and variational bounds

“…. are unknown constants, q ¼ À (Haftbaradaran and Shodja, 2009). Displacement field of an embedded couple stress multi-coated elliptic fiber with imperfect interfaces under remote anti-plane loading Suppose that an unbounded couple stress elastic solid containing an n-phase couple stress elastic cylindrical inhomogeneity with ellipse cross-section is subjected to the far-field shear stresses, r 1…”

“…It is worth mentioning that Mori-Tanaka method was previously employed in the framework of couple stress theory but for composites with simpler microstructural/nanostructural descriptions. For example, the extension has been incorporated in the study of the anti-plane properties of composites comprising of circular uncoated nano-fibers with interfacial damage (Shodja and Hashemian, 2019), and elliptic uncoated nano-fibers with perfect interface (Haftbaradaran and Shodja, 2009). Also, the method has been extended to the study of the inplane properties where the elliptic uncoated nano-fibers are either aligned or randomly oriented (Shodja and Alemi, 2017).…”

Variational bounds and overall shear modulus of nano-composites with interfacial damage in anti-plane couple stress elasticity

“…One of the convenient approaches used for the estimation of the effective properties of heterogeneous materials is the Mori–Tanaka method (Benveniste, 1987; Mori and Tanaka, 1973). This method has been utilized not only in the context of classical theory of elasticity, but also has been employed along with some higher order continuum theories to predict the overall behavior of composites containing inhomogeneities in the order of a few nanometers (Alemi and Shodja, 2018; Haftbaradaran and Shodja, 2009; Shodja and Alemi, 2017). Mori–Tanaka method will be extended to work in the framework of the current theoretical developments.…”